A sphere of radius R has a uniform volume charge density ρ. When the sphere rotates as a rigid o

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A sphere of radius R has a uniform volume charge density ρ....

A sphere of radius R has a uniform volume charge density $\rho .$ When the sphere rotates as a rigid object with angular speed $\omega$ about an axis through its center (Fig. P30.74), determine (a) the magnetic field at the center of the sphere and (b) the magnetic moment of the sphere.

Key Concepts

Rigid Body Rotation

Rigid body rotation is the motion where every point in an object undergoes a rotation about a common axis with the same angular velocity. This concept is central when analyzing moving charge distributions because it ensures that the relative positions of the charges remain constant, allowing a straightforward determination of their instantaneous linear velocities and, consequently, the resulting current density.

Magnetic Field Calculation

The calculation of the magnetic field resulting from a given current distribution often involves applying Ampère’s law or the Biot–Savart law. In the context of a rotating charge distribution, once the current density or the magnetic dipole moment is known, these laws facilitate the determination of the magnetic field at specific points, such as the center of rotation. This concept is fundamental in linking the motion of charges with observable magnetic phenomena.

Magnetic Dipole Moment

The magnetic dipole moment is a measure of the strength and orientation of a magnetic source, akin to a small bar magnet. For continuous current distributions, it is typically computed by integrating the contributions of local current densities over the volume, where each contribution is weighted by its position relative to the axis of rotation. The magnetic dipole moment provides a compact way to characterize the magnetic field produced at distances far from the source.

Uniform Volume Charge Distribution

This concept refers to a charge distribution where the charge density is constant throughout the volume of an object. In electrostatics and electromagnetism, such distributions are important because they simplify the calculation of electric fields, potentials, and, in dynamic settings, the contributions to current densities when the charged body is in motion.

Current Density from Charge Motion

When a uniformly charged object rotates, each moving charge contributes to a macroscopic current. The current density, defined as the product of the charge density and the local velocity of the charge, becomes a key quantity in calculating magnetic effects. This approach bridges the gap between static charge distributions and the generation of magnetic fields in dynamical systems.

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